GalRep-EllCrv · Galois Representations attached to Elliptic Curves

The project GalRep-EllCrv builds on Lorenzo Furio’s PhD thesis, where he achieved breakthroughs in the study of Galois representations attached to elliptic curves. This classical field has been central to major advances in number theory, including the proof of Fermat’s Last Theorem. The project addresses problems linked to Serre’s uniformity question, a long-standing challenge at the heart of the field.The research is structured into three directions:[A] Develop faster methods to compute integral points on non-split Cartan modular curves, correcting and extending previous results and pushing the limits of computational techniques.[B] Complete the classification of 7-adic Galois representations, a cornerstone of Mazur’s Program B, by resolving a specific genus-3 curve and creating new algorithms with broader applications.[C] Prove the first uniform bound on Serre’s constant, a measure of hidden symmetries in elliptic curves, with wide implications for Diophantine geometry.Beyond its scientific goals, the project will advance open science by releasing code and results openly, foster dissemination through publications, talks, and outreach, and promote societal reflection by integrating ethics into mathematical events. The applicant’s ongoing engagement with the Mathematical Olympiads will further inspire younger generations and encourage diversity in mathematics.The fellowship will be hosted at the University of Bordeaux, under the supervision of Pierre Parent, a world leader in the study of elliptic and modular curves. The combination of the applicant's recent breakthroughs with the host's expertise will provide an ideal environment for this ambitious project and position the applicant at the forefront of research in modern number theory.